Wave filter with lossy inductors and capacitors



8. LILJEBERG Aug. 26, 1969 v WAVE FILTER WITH LOSSY INDUCTORS ANDCAPACITORS Filed Jan. 18 1965 2 Sheets-Sheet l w 1 G w u 1 6 w a a T Z 6w 4 A 4 $1 ,1 4b 4% v M 77 a n p Z 4 a 5 K :3 m .4 7 1 R. 4: a "M M L 5%"W \\..M ZJQ-W X,A ym

I NVEN TOR. 5252" Alia/E3556 'JY'YZF VZY Aug. 26, 1969 L'LJEBERG3,464,034

WAVE FILTER WITH LOSSY INDUCTORS AND CAPACITORS Filed Jan. 18, 1965 2Sheets-Sheet 2 F R E QU E N C Y INVENTQR. 5557 412/; JZBEEG irro awyUnited States Patent @1 3,464,034 Patented Aug. 26, 1969 ABSTRACT OF THEDISCLOSURE An electrical wave filter which can be used as a lowpassfilter or a high pass filter or a band pass filter or a band eliminationfilter. The filter is formed from individual electric components of aquality much lower than heretofore usable with filters.

Background of the invention In the electronic arts it is often necessaryto suppress or eliminate voltages or currents having certain undesiredfrequencies, and at the same time, pass voltages or currents havingother frequencies. Electrical wave filters are used for this purpose.Electrical wave filters take many forms; however, the type of filterwith which I am concerned, is formed from a plurality of electricalcomponents such as inductors and capacitors.

There are four functions which my filters (as well as known filters) canperform. A filter performing the first of these functions is known as alow-pass filter and passes, with very little loss or attenuation,voltages or currents at all frequencies from zero up to some selectedmaximum or cut-off frequency, while suppressing or eliminating (i.e.,inserts very high loss or attenuation) voltages or currents at allfrequencies higher than the cut-off frequency.

A filter performing the second of these functions is known as ahigh-pass filter and performs effectively the reverse of the low-passaction. More particularly, a high pass filter passes, with very littleattenuation, voltages or currents at all frequencies higher than someselected minimum or cut-off frequency and suppresses or heavilyattenuates voltages or currents at all ferquencies below the cut-offfrequency.

A filter performing the third of these functions is known as a band-passfilter and passes voltages or currents at frequencies falling within arange of frequencies having selected maximum and minimum cut-01ffrequencies and, at the same time, suppresses voltages or currents atfrequencies which are either less than the minimum cut-off frequency, orare larger than the maximum cut-off frequency.

A filter performing the fourth of these functions is known as aband-elimination filter and has overlapping upper and lower cut-offpoints to suppress all voltages or currents at frequencies fallingbetween the two cut-off frequencies and at the same time passes allvoltages or currents having frequencies falling outside of the frequencyrange to be suppressed.

An ideal filter would pass all frequencies to be suppassed with no lossand would block all frequencies to be suppressed with infinitely highloss. Moreover, the frequency interval between frequencies to be passedand frequencies to be blocked would be extremely small or narrow.Finally, the impedance as measured at the input or the output of thefilter would often be matched to that portion of an electrical circuitcoupled either to the output or the input of the filter to preventreflection losses. Such an ideal filter could only be constructed ofpure reactances (both inductive and capacitive) because the presence ofelectrical resistance would introduce undesired losses or attenuation.

No such pure reactances exist. Hence, it is conventional to useelectrical components of high quality wherein the electrical resistancethereof is held to a minimum. For example, inductances or coils of suchquality have a high ratio of reactance to internal resistance, or stateddifferently are said to have high Q.

Summary of the invention I have invented a new electrical wave filterformed from a plurality of individual electrical components which can beused as a low-pass filter, or a high-pass filter, or a band-pass filteror a band elimination filter, which employs electrical components ofevery (such as less expensive) quality. Moreover, I have developed a newmethod for designing filters using such components of poorer quality.

Brief mathematical summary of the principles of my invention In thediscussion which follows, references to equations and figures refer tothe equations and figures numbered and described below:

Brief description of the drawings In the drawings:

[FIGURES 1-7 illustrate various circuit configurations as employed in mymethod.

FIGURE 8 is a graph showing the choice of the resonant frequencies.

FIGURES 9-11 are various circuits of low-pass filters in accordance withmy invention; and

FIGURE 12 is a graph showing the relationship between gain and frequencyfor the circuit of FIGURE 9.

FIGURE 13 is a graph showing the distribution of the losses in thenetwork.

Detailed description of preferred embodiment My method will now be setforth first illustrating the analysis and synthesis of laddertype andbridged T networks having capacitive tree branches.

A specific laddertype network or eventually a ladder bridged Tcombination, consisting of cascaded bridge Ts, where the bridge branchmay bridge two or several nodes, will be descirbed. The key idea is toarrange the network equations so that a differential operator appears assI in the diagonal of the circuit matrix, A. Therefore those branches,that are picked as maximal trees, have capacitive elements, and the restof the branches, the chord barnches, then consists of inductivebranches. This choice preserves the rank of the matrix, withoutintroducing constraints. Losses are introduced as series resistors inseries with the inductances, and conductances in parallel with thecapacitances. The latent roots of the matrix A now are the same as thetransfer poles.

A basic element, either inductive or capacitive, with current source andvoltage source is shown in FIG. 1. The voltage over the element is EEg,and the current flowing into the element from external sources is 1-1Here E and I are characterizing the element, I is a current generatorapplied over the element and B is a voltage generator.

Several of those elements are connected by a connecting operator ofincidence matrix,

l= l12+ l11 to form a filter, i.e., the currents converge from differentdirections, represented by the branches, to a node, where furthermore acurrent source is added. Thus the currents converging to each node isIIZ b'l IH c lm ob-l- 111 0 Premultiply by 6 b+ l12 ll1 o 0b+ lli lllflu Similarly for the chord matrix is obtained o-( l12 l11)' b= oc( liz'lnl' ob The key idea now is to choose the elements so that s=d/dtappears only once in each equation, preferably in the main diagonal ofthe matrix of the system of equations. Thus setting YE =I and ZI =E thetwo equations are In the matrix Equation 1 the basis or independentvariables are the components of the vector or set [l E Here the index brefers to the capacitive tree branches Thus, E is the set of voltagesover the capacitances Y, of the network, and I is the set of currentsthrough the coils, Z. The index 0 refers to the inductive chordbranches, Z.

Dependent variables in Equation 1 are all the current sources andvoltage sources, 1 E applied to each branch element, see FIGS. 1 and 2.In the case of an active network, part of the sources may be considereda function of some of the independent variables, e.g., 13,; of a vacuumtube, and must be included in the left-hand member of the array 1. Asfor independent variables, index b refers to tree branches and index 0to chord branches. Thus Equation 1 is a transformation from E 1 to E IApplying to the network E 1 E and T can be solved for by premultiplyingboth sides by the inverse of the left-hand matrix, if that matrix hasrank =n= same as the order of the matrix.

The resonant-frequency matrix is contained in Equation 1.k. Theresonant-frequency matrix is a matrix with the coupling terms asindependent variables and the loopresonant frequencies as dependentvariables in such a way that the coupling terms are obtained as thesolution of the resonant-frequency matrix. This solution is iterated toa set of closer to correct values of the coupling terms. In one of theiteration steps the coupling terms,

is obtained, where y y, y,, are the square of the resonant frequenciesof the loops of the prototype ladder-filter, and f(y) is the correctedeven part of the transfer polynomial, corrected by the r-terms, theloopresonant frequencies, the coupling, and the half-power bandwidthsobtained in the previous step of the iteration (.Newtons method).Obviously for zero coupling the resonant frequencies and the bandwidthsof the isolated loops coincide with those of the tarsnfer poles.Therefore a suitable initial choice of the loop-resonant frequencies isa choice close to the corresponding frequencies of the transfer poles insuch a way that the coupling is positive or realizable. Another suitablechoice in the initial steps of the interation is a choice of the rippleratio, r (containing the b-terms as functions of q close to zero. Theripple in the passband is then changed to the right amount by increasingr so that the choices of the entries of the circuit matrix arerealizable as physical circuit elements.

In the eighth order case the equation are 1.1, 1.2, 1.3, and 1.4. Theseequations can be solved for s s s s and S5S7 as functions of theremainder of the entries of the circuit matrix such as the loop-resonantfrequencies, s +s s +s s +s and s r, and the r-correction terms,containing frequencies, s and half-power bandwidths, q.

In the fourth order case the equations are 21, and 23. The solution ofthe coupling term, x in the final step of the iteration as a function ofthe resonant frequency, x;,, of either of the loops is shown in FIG. 8.The crossings with the x -axis, .622 and 2.412, i.e. the correctedvalues of the transfer poles, are close to the initial choice, .414 and2.414, i.e. where the ripple ratio, r or t, was chosen equal to zero.The choices of the two resonant frequencies, x and x and the resultingcoupling, x is shown by the dotted line in FIG. 8. For the maximum valueof the coupling, x the two resonant frequencies are equal, x =x 1.225,and the characteristic impedances of the two links are better matchedbecause of the closer coupling.

The loss-distribution matrix is contained in Equation 2.k. Theloss-distribution matrix is a matrix with the coupling-loss terms asindependent variables and the losses of the loop circuits plus ther-correction terms as dependent variables in such a way that thecoupling loss terms are obtained as the solution of theloss-distribution matrix. The losses of the loop circuits are the sum ofq +q where i is odd, and they are a measure, as usual, of the quality orQ-values of the loop-resonant circuits, with a loop-resonant frequency,

i odd. The coupling losses are the sum of qj+qj+i where j is even, andthey are associated with the coupling terms, s (j even), and they arecomprised of one part from each of two consecutive loop-resonantcircuits. The solution of the loss-distribution matrix serves as aniteration step in the procedure of finding realizable entries of thecricuit matrix. The solution of the above frequency matrix is preferablyfollowed alternately by the solution of the loss-distribution matrix. Asuitable initial choice of the Q-values is a choice close to the valuegiven by the real part of the corresponding tarnsfer poles, or a choiceof the ripple ratio, r, close to zero. The Q-values should be betterthan given by the transfer poles, if the coupling losses are going to berealizable. However, negative bandwidths are permissible, because theycan be realized by means of negative resistors, in the case where thesolution of the loss-distribution matrix is negative.

In the eighth order case the equations are 2.1, 2.2, 2.3, and 2.4. Theseequations can be solved for q +q q +q and q +q in terms of the entriesof the circuit matrix such as r, the Q-values of the loop-resonantcircuits, the resonant frequencies (in the r-correction terms), and thetransfer poles of the filter. In the fourth order case the equations are20, and 22. The solution of the coupling loss in the initial step of theiteration for r=t=0 in terms of the half-power bandwidth of either oneof the loops is shown in FIG. 13. With an initial choice of the resonantfrequency corresponding to s =.75 and s =1/s ==1.333 the coupling lossmust lie on the line between the lines for s =.7 and -.8 in FIG. 13. Anincrease of the Q-value of s corresponds to an increase of the couplingloss. For passive filters the choices must be within the non-shadedrealizability region (positive q,). The region on top of thecircleshaped area is realizable by negative coupling, x yieldingnegative capacitor, C On the verical line, where the coupling is zero, s=2.414 with bandwidth 2, and s =.414 with bandwidth 4.28. The regioninside the circleshaped area is non-realizable. The line parallel to theabscissa with s =s corresponds to the matched filter above and yieldszero q and g Thus this case can only be realized by means of a losslessfilter terminated in an input or output resistor. The realizabilityboundaries are slightly changed in the final iteration steps. However,if the initial choices are made well in the center of the realizabilityregion, the final filter is most likely realizable.

The invention is not necessarily restricted to the exact embodiment ofthe lossy ladder filter herein described and illustrated but thatmodifications such as the choice of the matrices Z and Y as transformer,or with diagonal entries of the form a ls+a or 1/ [w s+a may be made bypersons skilled in the art of application of matrix algebra toengineering problems such as the reduction of resulting matrix to thetridiagonal prototype lossy-ladder form and that all such embodimentsare considered to fall within the scope of this invention. Then therealizability region above may be extended to negative (but not complex)values, because the s and the q, of the ladder become functions of theoff-tridiagonal entries of the circuit matrix.

The entries of the matrix Equation 1 consist of the admittances and theimpedances of the network and the interconnecting operator.

Each of the maximal tree branches consist of an admittance or Y =sC +GThe set of these maximal tree branches, which in FIG. 3 are drawn fromthe reference or ground terminal 0, is in Equation 1 written as asubmatrix with the capacitive elements as entries Y. This matrix Y canalso represent a capacitive coupling network.

Likewise each of the chord branches consists of an impedance or Z =sL +RThe set of these chord branches, in FIG. 3, is in Equation 1 written asthe submatrix, Z with entries (2;). The matrix Z can also be atransformer or a rotating-transformer matrix.

The interconnecting operator is 112 111 which is equal to 6 the lastpart of the incidence matrix in the case of one capacitive tree branchto each node, for the ladder-type network considered. Theinterconnecting operator has only +1 or -1 entries if the network is passive, which entries are dimensionless. In case of feedback it may alsocontain nonlinear amplification factors or transconductances. In theladder type network there are only three connections at each node. Theinterconnecting operator therefore contains at most two entries in eachrow and each column, representing the two chord coils Z and Z at node 7,FIG. 3, together with a diagonal entry from the matrices Y and Z.

The system Equation 1 rewritten in matrix notation where p is the numberof capacitances and q is the number of inductances, yields the matrixrelation c. (AJFSDLT. #5.]

S c. 0 v6;

own] nzHE. t F,

[B -PSI S 6 where (1 0 0 Q2 0 O 0 0 0 0 De U. F".

O 0 q2p1 0 0 qzq f12 0 0 f12q Originally in 1950 the circuit matrix wasobtained in the form of a superdiagonal and a subdiagonal of resonantfrequencies and the same principal diagonal from Zobel filter chainconsiderations of the input admittance. The 3 db bandwidth of thecapacity C is q =G /C w due to losses introduced by the conductance G inparallel with the capacity, and q =R /L is proportional to the 3 dbbandwidth of the coil L due to losses from the resistor R in series withthe coil. The pxq matrix S contains entries proportional to the resonantfrequencies of various combinations of coils and capacitances. In caseof a ladder type network, S is a triangular matrix with a principaldiagonal and an auxiliary subdiagonal. After performing the samepermutation on the rows and the columns of A the submatrix S can beobtained from the auxiliary superdiagonal of A. As q q and s have thesame proportionality factor this may be taken out of the matrix as ascalar multiplier, or they may be just the bandwidths respectiveresonant frequencies, e.g., measured with respect to a referencefrequency f =1.

The system matrix A can be changed in form by various collineatory orsimilar transformations B=P* AP in order to simplify A, e.g. to itstridiagonal form, or

in order to bring A over in some of its canonical forms, e.g., thecanonical form of its characteristic roots D=X* AX Such a similartransformation leaves the characteristic roots invariant, i.e., Equation10 is X* (A+sI)X=X AX+X* sIX-=D+sl The system of equations representedby the network may be written low/E x= X m Ci a/E here x is a newindependent variable and y is the new set of applied voltage and currentsources. This new network equation is a set of independent differentialequations, that can be solved one at a time. Thus the first one is withthe solution x -=A /t/e t where A /t/ is the constant of integration.The magnitude A /t/ is completely independent of x x x Other outputs maybe obtained by summation of the x s, i.e., other outputs depending oninitial conditions. Thus the output Eta 5i A /t/e 1 i IBVITi A /t/e(a+iw )t The transfer poles s a -l-ja i=1, 2, n are thus the same as thecharacteristic roots of the matrix A, with the arrangement of .91 in thediagonal of the matrix Equation 1. The transfer poles and thecharacteristic roots of A are also roots of the circuit determinantdet(A +sI)=det(D+sI)'=0 a polynomial in .1, which is the same no matterhow the circuit equations are obtained. The transformation X is the setof characteristic vectors, a solution of the equation where y is the oldvariable and x is, as above, the new independent variable and H is thesimilar transformation characterized by HH=L A is changed to asymmetrical matrix, with entries in the complex field for a networkwithout controlled sources.

Thus the matrix A in Equation 10 is premultiplied by E andpostmultiplied by H in the first part of Equation 1d to obtain the newcircuit matrix A with an symmetrical imaginary part with positive sign.In the second part of Equation 1d A is premultiplied by H andpostmultiplied by 'H to obtain the equation with negative symmetricalimaginary part. The operation is to introduce I=HII between A and y inany similar transformation:

AIy=AHHy,

and furthermore multiply each equation by E to obtain fiAHfiy. Here H isunitary.

Example For example Equation 1 for the network in FIG. 4 (shown indifferent form in FIG. 5) is:

According to Kirchhoffs current law: Current flowing from node 1 is (sC+G )E I -I and current flowing to node 3 is -I +(sC +G )E I =0. This isthe incidence of the currents at the discontinuous points 1 and 3.According to Kirchhofis voltage law: Voltage around mesh 013 is:

and voltage around mesh 035 is a ed- 4) 4+ o5= and voltage around mesh015 is giving the circuit chord matrix from the sum of the voltagesaround the meshpath.

Taking out the independent variables from the array Equation 1 above isBy adding one more link, the network in FIG. 6 can be obtained.

Similarly the matrix Equation 1 becomes Y -1 -1 -1 E1 -Y3 -1-1 -1 3 1 s0 E07 1 1 Z 1 1 Z 1 1 e 11 17 01 Observe that the element 0 is ofopposite sign as compared to the other entries in the submatrix in theupper right hand corner. Because of skew symmetry 0 is equal to -cCompare zero of transfer function on imaginary axis.

A more general appearance of this special matrix in Equation 1 is whichgives an outline of how S looks like also in the nth order case. Everysecond of the subparts of S has a plus sign in the diagonal that is ofone lower degree than the other.

Omitting I R in Equation 1 in the example, and premultiplying by IN? andpostmultiplying by the same, gives in the field of real numbers 'iil 'i/V 2 1 in the field of complex numbers.

My next example illustrates the synthesis of a fourdimensional matrixfor a low pass ladder type network.

Any matrix satisfies its characteristic equation according to theHamilton-Cayley equation. As there are n elements in the characteristicequation when A is substituted, it is possible to set up n equations inorder to determine the entries of A. However, as there is not more thann characteristic roots, not more than n circuit elements, including oneresistor, has to be considered in the synthesis. The other circuitelements are more or less arbitrary, in choice, so far as they arepositive entries.

The characteristic equation is det/ A +sI/ 0 Substituting A instead of sgives for the characteristic equation.

A comparison of the equation det(A+sl) and (s) both equal to zero, givesfor the coeflicient of s" as the imaginary parts of the roots cancel.

The same result is also obtained by setting the element in the lowerleft hand corner of the characteristic matrix equation=0, or by settingthe element in the upper right hand corner=0.

In the same way n kid, llsk=the sum of the diagonal elements in A.

I now proceed to the synthesis of a 4-dirnensional ladder type matrixwith single-ended matching resistor and transfer poles s"=1-.Lj2. ands"=-2ij, the resulting low pass ladder type network.

The matrix is The sum of the coefiicients of s gives Substituting thisvalue of q in A the Equation 1e be- Set the first column in Equation1:0. The entry with index 31 is and 11 is (29:13) (17-13)+13s 8 13:0Thus s /40/ 13 and s /25/ 13.

The preceding synthesis can be modified for a 4- dimcnsional laddertypematrix with input and output matching resistor and transfer poless'==-1: 1'2 and s"=2:j, as described below.

The sum of the coefiicients of .9 gives, if both matching resistors areequal As the matrix is symmetrical about the secondary diagonal, theentry s is equal to the entry s and .9 equals s42, as indicated in Aabove. Because the characteristic roots are the same as in the previousexample, substitute A in Equation 1 of the previous example.

Thus entry 31 set=0 gives 3s 3=0, .9 1, s 1, and entry 21:0: s=2, and

An Involutorial Automorphism of the Scalars of a Vectorspace onto theScalars of the Dual Vectorspace.

Given the vectorspaoe, V, over the division ring, A, and the dual space,V*, over A of a linear transformation, T, with matrix, A, the elementsof A, according to Jacobson, are anti-automorphic to A, if A possessesan anti-automorphism. In the case of a field this is an automorphism.

The automorphism of a sza l ar c=a+kb in V is defined c*=a-kb Even c=a,Odd c=b where c, keK, a, b, k eF, e.g. the field of R# or C#.

The norm of c is defined because lim R is never exactly equal to zero. a

A vector in V is x =(a ,+kb and therefore a vector in V* is x =(a ,kb)=(c i, j, r=1, n with scalar components in K. Thus the scalar productof the two vectors is 11 i) E i: i:

Furthermore, the scalar products are Hermitian in the sense that 1 j)jr 1) With the map i k R# F the theory may be developed by means ofsimilar operations as in the Hermitian case.

The set of transformations, T with matrix A=[a, considered in this papercan be synthesized with the above type of scalar products as entries.

Consider the case of the similarity transformation of A to the diagonalmatrix of its characteristic values, A. Here the basis of V isorthogonal in the sense that where x is an eigenvector obtained from(A-)\ l)x =0 If X [x,,] with the eigenvectors x,- as columns, thisimplies that In the above Equation 2 set r=s. The last two members inthe sum vanish identically for all i=1, n, because of the commutativeproperty of the multiplicative group of the field F. If A, were a memberof K, the condition of symmetry of the scalar products in the Hermitiansense would require that k vanish in the first two members of the sum in(2). As a, b in general are arbitrary members of F, cannot contain k.

Theorem: The elements on the principal diagonal of A and thecharacteristic values of A belong to the field F.

Furthermore, the Hermitian condition requires that the matrix A has asymmetric even part and a skew-symmetric odd part. In (2) replace r by sand use the commutative property of F. The even part of a remains thesame as in a but the odd part changes sign. Thus A=XX* =I 11 n n 2 ri eii i i i i i i) i.e., elements on both sides of the principal diagonalare equated to zero.

11. E ri -i ri ai i= Example: The matrix [l-i3 2-ki2 2+ki2 1+i3 has thecharacteristic equation with the roots A1 z=1ii The first eigenvectorobtained from (A)\ I)x =0 1 [1,1+k 1 1 1=-1 Thus the first normalorthogonal eigenvector is For the second eigenvector choose NormalizeNormalize Le, the second eigenvector is -1 k i 1 Reduce A to itsdiagonal form X* AX=A by the orthogonal transformation of itscharacteristic vectors.

Thus, in accordance with my method, a general linear network, system, orstructure can be synthesized in the following two steps:

(1) All n or less relations between the entries as functions of thecircuit elements of the network (usually known as the lumped constantsof the network) and the natural frequencies (transfer poles), orcharacteristic values of the circuit matrix A of order n must beestablished, which relations leave the above characteristic valuesinvariant.

(2) Those n or less relations are solved by elimination for n or less ofthe unknown circuit elements expressed as a function of the naturalfrequencies and the remaining circuit elements more or less arbitrarilychosen within bounds given by the existence of the invariants of thesystem of relations, such that said circuit elements become realizablefrom a physical point of view such as positive or negative resistors,positive inductors and capacitors.

Hence, the objective of my synthesis procedure is to find a set ofvalues of the inductors L,, the capacitances C and the resistorsR, and6,, also expressible as functions of the entries s, and :1 of thecontinuant matrix of the prototype ladder network. The response or thecharacteristics of the network is given in the nth order differentialequation, which has been analyzed previously with known methods, andwhich for example can be factored into its natural frequencies, rcqi 116From the relationship between the s (1,, 8 and 04 the entries of thecontinuant matrix are calculated. After that the circuit elements, L C Rand G with characteristics indicated by the calculations are determined,made and connected together according to the specifications contained inthe incidence matrix or the drawing of the circuit, network, simulatoror structure, which it is desired to design or synthesize. The synthesisprocedure embodies the following n relations between the above matrixentries, .9, and q,, the last of which b, is a symmetric function, thenatural frequency components, 5, and on the last of which a is asymmetric function, and a set of matrix entries, s and q satisfying then relations, corresponding to physically realizable inductors,capacitors and resistors, L C R and G or their analogies. The logicaloperations on the indices, 2', 1', and k, which occur in the relations,can suitably be programmed on a digital computer. For example, Newtonsmethod can be used to solve for any n unknowns from the set, {s q,}, interms of the remaining n-l s,, q; and the 3 (1 If r is small, which forexample, is the case for passive and moderately active filters, the setof equations (l.k), which are of even weight in the frequencies s and qcan be approximately solved for n/2 (12 even) or (n1)/2 (n odd) of thelast or first resonant frequencies, s Likewise, the second set ofequations, (2.k), which are of odd weight in the s and the q,, can besolved for n/ 2 pairs of the half power bandwidths, q +q (n even), or(n+1)/2 of the half power bandwidths, q;, (n odd).

The symmetric type function, 8 is a sum of products of variousquantities associated with the circuit elements of the branches, asindicated by the indices of the quantities. An index i in q, wouldindicate that it is associated with the i' branch. The logicalproperties of the product members of the 8;, can be accounted for in theindex rules:

(1) The index i in q, counts as i. (It is associated with the propertiesof the circuit element in the i branch.)

(2) The index i in s, counts as i and i+1. (It is associated with theresonant frequency s =j/ /L C +1, between the bran-ches i and i+1 in theladdertype case. Also, in the matrix, it is the entry in the i rowand'the (i-i-l) column.)

(3) An index enters only once in each product member of the S (4) Allindices 1, 2, 3, n occur in a product member of the S In thetimeinvariant case, the symmetric type functions, S in Equations l.k and2.k are In Equation 3 the symmetric function, bj, is the sum of allcombinations of products of a subset of the half power bandwidths, q,,taken j at a time. The indices of the physical quantity, q,, in thissubset belong to the subset of the n indices, which is the complement ofthe indices associated with the s, in the same product member of S -Inparticular b =1, and, if (D7 is in this subset of indices,

and

0( ia j) j Denoting the natural frequencies n for odd n the symmetrictype function, 8,419,, as), in Equation 4 is obtained, in a similar way,from the S (s p in Equation 3 by setting all s s equal to zero, and bysetting (which also means that the value b is replaced by the value a;)

of the amount of ripple in the passband of the sinusoidal response ofthe filter. Also, r is the ratio in large of the real part, or to theimaginary part, ,3 of the natural frequencies in the s-plane (Equation=6), and the corresponding ratio of the half-power bandwidths, q,, tothe resonant frequencies, s In filter design, where circuit elementswith relatively small losses are used, r is a small quantity, forexample if the largest 04 is set equal to one, in a filter withChebyshev approximation with 30% ripple in the passband, r isapproximately equal to 1.8/n. If r is close to zero, the filter islossless of combtype with a delta-function response, 6 (to), at thebreak frequencies, 8;. In this case of small r the coefficients of thehigher degree terms in r in the n relations. Equations 1.k and 2.k, canbe considered a relatively small correction to the symmetric functionsof p, and :1 Therefore, a good first approximation is obtainable fromthe study of a lossless LC filter according to Equation 1.k, and fromthe determination of the loss distribution of this LC filter accordingto Equation 2.k, with r close to zero or an estimation of thecorrections from the r terms.

cording to the above index rules are 1 2 3 4 6 8 7 51 1 3 35 51 1 3 4'i' 5 a 'l' 1 Diagonal multiplication rules of i=1, 2, n

The order of the product=the sum of the orders of the components. (I)+jth order matrix times a second matrix of +kth order yields an increaseof i of the second matrix by j.

1 -1)(1 0 1+ o+ -1 ad- 3+ (jzk as only the principal diagonal andsuperdiagonals are considered).

W 8}]0080 (1 [0000 000 1 b [0000 aim-H 0] (III) kth order matrix times asecond matrix of +jth order (jzk) yields a decrease of both indicesby+k.

y 1st element 0 O\ ab- The first element should be a b but as i 1, forall k the new index becomes zero or negative and those 20 indices do notexist as i=1, 2, n. Therefore the first entry is a b 1=order i i+l+ iil+ il+ i il+ i i+l 21 22= i i+l+ i il+ il+ i il'i' i iH Second order1': 1, 2

Add to the last equation the product of (1 1) i 2+ i -1 i i +1+ i +1+ 94and 1 a (qs-q1)(qrQ1)]=[(q 1) +fl1 ][(q1+ 2) +B2 Normalize the s i=1, 2,3 by dividing by [3 5 set i f. X1 132 X2 5132 and and set q1=q2= s=qThen the four relations (20), (21), (22) and (23) are q+q4+ 1+ z=0 (30)X is solved for from Equation 33 where X can be chosen such that X isrealizable. Substitute X into Equation 31 and solve for X Here X can bechosen (with a known q) such that X becomes positive or realizable. SeeFIGURE 8.

q; is solved for from Equation 30 Finally, q is determined fromessentially Equation 32, see FIG. 13. Substitute X X and (1 intoEquation 32 Equations 27, 28, 29, and become or t'-'=% for a 2.5%ripple, or t .=.053 for 30%.

I now show the application of my method using the Chebyshevapproximation with 2.5% ripple in the passband (see FIG. 8).

The attenuation is d =vm where a =.025 for a 2.5 ripple in the passbaud0: cos- :0 is a function of frequency If 11 :0, 0 is obtained from cosh86=1/a =l/.025=40, :4.40 0=4.40/8=.55 tanh 0=tanh .55=.50052=.5 t'-=tanh0= A and plotted in FIGURE 8. x maximum occurs at dx /dx =:=1+1.5/x x1.5 1.225 and is x =3.03553-2.45=.58553 The crossings with the x -axis)are With x =.75, x is obtained from Equation 28 26 A suitable choice ofx;; is .75, which is in the realizability region, and displaced withrespect to the maximum of x such that a relatively large q can berealized. The value of q is calculated from Equation 27 It is convenientto use Newtons method of approximation to find a value of the quantity(1q), which satisfies this equation.

h= .0044/1.99= .00221; (1 q) .O6221E.062

h=.000679/1.98=.000342; (1q)=.06234 q=.93766 FIGURES 9-11 illustratevarious low pass filters designed in accordance with my method and usinginductors having relatively high resistances and capacitors havingrelatively low losses. Each inductor is identified as L with anappropriate lower case number, the resistance of each inductor beingidentified as R with the same lower case number and shown in series withthis inductor. Each capacitor is identified as C wtih an appropriatelower case number, the resistive lossy component of each capacitor beingidentified as G with the same lower case number and shown in parallelwith this inductor.

As shown in FIGURE 9, two inductors L and L together with their seriesconnected resistors R and R are connected in series between inputterminal and output terminal 30. Input terminal 22 is connected directlyto output terminal 32. The junction of the two inductors is connected tothe junction of terminals 22 and 32 by capacitor C and its paralleledloss G Similarly, capacitor C with its paralled loss G is connectedacross output terminals and 32.

FIGURE 10 extends the circuit of FIGURE 9 to the use of three capacitorsC C and C each with a corresponding resistance R R and R respectively,and to the use of three capacitors C C and C each with a correspondingresistance loss G G and G respectively.

FIGURE 11 generalizes the application of FIGURE 10 to an indefinitelylarge even number n of components where the number of inductors is n/ 2(each inductor being identified by an even number, i.e., 2, 4, n) andthe number of capacitors is also n/2 (each capacitor being identified byan odd number, i.e., 1,3, (n1).

The circuit of FIGURE 9 was designed as a low pass filter in the radiofrequency range, to start increasing attenuation at about 450 kilocyclesper second and to have an effectively uniform low attenuation atfrequencies below thisvalue. More particularly when the values shownbelow are used, the relative gain as a function of he quency was foundto meet the above requirements.

Circuit values L millihenrys 0.511 L do 0.191 R ohrns 264 R do 419 Cmicromicrofarads 330 C do 2425 G ohms 5700 G do 790 The resulting curveof frequency vs. relative gain using test data on the circuit of FIGURE9 using the above values, is shown in FIGURE 12. Note the extremelyclose correlation between theory and test data.

While certain novel features of my invention have been shown anddescribed and are pointed out in the annexed claim, it will beunderstood that various omissions substitutions and changes in the formsand details of the device illustrated and in its operation can be madeby those skilled in the art without departing from the spirit of theinvention.

Having thus described my invention, I claim as new and desire to secureby Letters Patent:

1. A network comprising:

first, second, third and fourth terminals;

a first series circuit connected between the first and second terminalsand consisting of a first voltage source, a first resistance and a firstinductance in series connection;

a second series circuit connected between said second and thirdterminals and consisting of asecond resistance and a second inductancein series connection;

a third series circuit connected between said first and third terminalsand consisting of a third resistance and a third inductance in seriesconnection;

a second voltage source connected between said first and fourthterminals;

a first capacitor connected between said third and fourth terminals;

a first conductance shunting said first capacitor;

a second capacitor connected between said second and fourth terminals;and

a second conductance shorting said second capacitor.

References Cited UNITED STATES PATENTS 2,922,128 1/ 1960 Weinberg 333-742,760,167 8/1956 Hester 333- 2,029,014 1/1936 Bode 333-70 1,958,7425/1934 Cauer 333-70 2,342,638 2/ 1944 Bode 333--70 OTHER REFERENCES W.H. Chen: Linear Network Synthesis, pub. by McGraw-Hill, 1964, p. 346-53.

L. Weinberg: Synthesis of Unbalanced LCR Networks, Journal of AppliedPhysics, vol. 24. pp. 300-306, 1953.

HERMAN KARL SAALBACH, Primary Examiner C. BARAFF, Assistant Examiner US.Cl. X.R. 333-74, 75

